Most primitive groups are full automorphism groups of edge-transitive hypergraphs

We prove that, for a primitive permutation group G acting on a set of size n, other than the alternating group, the probability that Aut(X,Y^G) = G for a random subset Y of X, tends to 1 as n tends to infinity. So the property of the title holds for all primitive groups except the alternating groups and finitely many others. This answers a question of M. Klin. Moreover, we give an upper bound n^{1/2+\epsilon} for the minimum size of the edges in such a hypergraph. This is essentially best possible.Comment: To appear in special issue of Journal of Algebra in memory of Akos Seres

On The Robustness of a Neural Network

With the development of neural networks based machine learning and their usage in mission critical applications, voices are rising against the \textit{black box} aspect of neural networks as it becomes crucial to understand their limits and capabilities. With the rise of neuromorphic hardware, it is even more critical to understand how a neural network, as a distributed system, tolerates the failures of its computing nodes, neurons, and its communication channels, synapses. Experimentally assessing the robustness of neural networks involves the quixotic venture of testing all the possible failures, on all the possible inputs, which ultimately hits a combinatorial explosion for the first, and the impossibility to gather all the possible inputs for the second. In this paper, we prove an upper bound on the expected error of the output when a subset of neurons crashes. This bound involves dependencies on the network parameters that can be seen as being too pessimistic in the average case. It involves a polynomial dependency on the Lipschitz coefficient of the neurons activation function, and an exponential dependency on the depth of the layer where a failure occurs. We back up our theoretical results with experiments illustrating the extent to which our prediction matches the dependencies between the network parameters and robustness. Our results show that the robustness of neural networks to the average crash can be estimated without the need to neither test the network on all failure configurations, nor access the training set used to train the network, both of which are practically impossible requirements.Comment: 36th IEEE International Symposium on Reliable Distributed Systems 26 - 29 September 2017. Hong Kong, Chin

Angular behavior of the absorption limit in thin film silicon solar cells

We investigate the angular behavior of the upper bound of absorption provided by the guided modes in thin film solar cells. We show that the 4n^2 limit can be potentially exceeded in a wide angular and wavelength range using two-dimensional periodic thin film structures. Two models are used to estimate the absorption enhancement; in the first one, we apply the periodicity condition along the thickness of the thin film structure but in the second one, we consider imperfect confinement of the wave to the device. To extract the guided modes, we use an automatized procedure which is established in this work. Through examples, we show that from the optical point of view, thin film structures have a high potential to be improved by changing their shape. Also, we discuss the nature of different optical resonances which can be potentially used to enhance light trapping in the solar cell. We investigate the two different polarization directions for one-dimensional gratings and we show that the transverse magnetic polarization can provide higher values of absorption enhancement. We also propose a way to reduce the angular dependence of the solar cell efficiency by the appropriate choice of periodic pattern. Finally, to get more practical values for the absorption enhancement, we consider the effect of parasitic loss which can significantly reduce the enhancement factor

Core Decomposition in Multilayer Networks: Theory, Algorithms, and Applications

Multilayer networks are a powerful paradigm to model complex systems, where multiple relations occur between the same entities. Despite the keen interest in a variety of tasks, algorithms, and analyses in this type of network, the problem of extracting dense subgraphs has remained largely unexplored so far. In this work we study the problem of core decomposition of a multilayer network. The multilayer context is much challenging as no total order exists among multilayer cores; rather, they form a lattice whose size is exponential in the number of layers. In this setting we devise three algorithms which differ in the way they visit the core lattice and in their pruning techniques. We then move a step forward and study the problem of extracting the inner-most (also known as maximal) cores, i.e., the cores that are not dominated by any other core in terms of their core index in all the layers. Inner-most cores are typically orders of magnitude less than all the cores. Motivated by this, we devise an algorithm that effectively exploits the maximality property and extracts inner-most cores directly, without first computing a complete decomposition. Finally, we showcase the multilayer core-decomposition tool in a variety of scenarios and problems. We start by considering the problem of densest-subgraph extraction in multilayer networks. We introduce a definition of multilayer densest subgraph that trades-off between high density and number of layers in which the high density holds, and exploit multilayer core decomposition to approximate this problem with quality guarantees. As further applications, we show how to utilize multilayer core decomposition to speed-up the extraction of frequent cross-graph quasi-cliques and to generalize the community-search problem to the multilayer setting